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h1. Δv Diagram

{excerpt}A graphical approach to understanding the form of the centripetal acceleration.{excerpt}

h3. Assumptions

We assume that we have _uniform_ circular motion (motion with a constant radius and a constant speed centered at a fixed point in space).

h3. The Diagram

The picture below illustrates the motion, with coordinates chosen so that the angular position at _t_ = 0 is θ = 0.  

!DeltaV.png|width=600500!

To the right of the motion diagram is a vector diagram that shows the change in the velocity vector.  The picture motivates the conclusion that if we take a very small Δ_t_, the change in the velocity approaches:

{latex}\begin{large}\[ \Delta\vec{v} \rightarrow - v(\Delta \theta)\hat{r}\]\end{large}{latex}

In the infinitesimal limit, this equation becomes:

{latex}\begin{large}\[ \frac{d\vec{v}}{dt} = - v \frac{d\theta}{dt} \hat{r}\]\end{large}{latex}

Using the fact that for _uniform_ circular motion,

{latex}\begin{large}\[ \frac{d\theta}{dt} = \frac{v}{r}\]\end{large}{latex}

we arrive at the form of the centripetal acceleration:

{latex}\begin{large}\[ \vec{a} = \frac{d\vec{v}}{dt}= -\frac{v^{2}}{r} \hat{r}\]\end{large}{latex}

h3. Analogy with Gyroscopic Precession

Consider a gyroscope precessing.  The angular momentum will trace out a circle as shown below.

!DeltaL.png|width=500!

The similarity to the Δ_v_ diagram implies that we can write:

{latex}\begin{large}\[ \frac{d\vec{L}}{dt} = L\frac{d\phi}{dt}\hat{\phi} \]\end{large}{latex}

where the derivative of φ with respect to time is the angular frequency of precession, usually written as Ω:

{latex}\begin{large}\[ \frac{d\phi}{dt} = \Omega\]\end{large}{latex}

h4. Fundamental Relationship for Gyroscopes

With that substitution, we have arrived at the fundamental relation for gyroscopes:

{latex}\begin{large}\[ \frac{d\vec{L}}{dt} = L\Omega\hat{\phi} \]\end{large}{latex}

{note}It is important to note that the relevant model for gyroscopes, [Angular Momentum and External Torque], contains the Law of Change:
\\
{latex}\begin{large}\[ \sum \vec{\tau} = \frac{d\vec{L}}{dt} \]\end{large}{latex}
\\
Often, this Law will have to be substituted into the fundamental relation for gyroscopes.{note}


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